QTA 5 - SAMPLE MOMENTS Flashcards
What is the formula for estimating the mean?
πΜ = (1/n) β Xπ for i=1 to n
The mean estimator is based on independent and identically distributed random variables
What is the difference between an estimator and an estimate?
An estimator is a function that calculates an estimate based on data, while an estimate is the value produced by applying the estimator to data
What does the bias of an estimator measure?
The bias measures the difference between the expected value of the estimator and the true population value
What does it mean that the mean estimator is BLUE?
BLUE stands for Best Linear Unbiased Estimator, indicating it has the lowest variance among linear unbiased estimators
Define consistency of an estimator.
An estimator is consistent if it converges in probability to the true value as the sample size increases
How do the Law of Large Numbers (LLN) and Central Limit Theorem (CLT) apply to the sample mean?
LLN states that the sample mean converges to the population mean as sample size increases, while CLT states that the distribution of the sample mean approaches a normal distribution as sample size increases
What are skewness and kurtosis used to estimate?
Skewness measures the asymmetry of a distribution, while kurtosis measures the tail heaviness of the distribution
What is the formula for estimating the variance?
π^2 = (1/n) β (Xπ - πΜ)Β² for i=1 to n
The sample variance is a biased estimator
What is the bias of the sample variance?
The bias is E[π^2] - πΒ² = -πΒ²/n
The sample variance tends to underestimate the population variance
What is the relationship between standard deviation and standard error?
Standard deviation measures uncertainty in a data set, while standard error measures uncertainty of an estimator and decreases with increasing sample size
How are annualized means and standard deviations calculated?
Annualized mean = monthly mean Γ 12, weekly mean Γ 52, daily mean Γ number of trading days per year
Standard deviations are scaled using the square root of the same factors
What is the significance of log returns in finance?
Log returns allow for the summation of consecutive returns and are convenient for analysis
What is the impact of sampling frequency on skewness?
Skewness does not follow a simple scaling law across sampling frequencies and may change direction at different frequencies
What defines excess kurtosis in financial data?
Excess kurtosis indicates that the data has heavier tails than a normal distribution, typically seen in financial returns
What are histograms used for?
Histograms represent the frequency distribution of a data series by dividing the data range into bins and counting occurrences
How does a kernel density plot differ from a histogram?
A kernel density plot uses a weighted count of observations to provide a smooth estimate of the distribution, unlike the discrete bins of a histogram
What is a linear estimator of the mean expressed as?
πΜ = β π€πXπ for i=1 to n, where π€π are weights that do not depend on Xπ
What does BLUE stand for in statistical estimation?
Best Linear Unbiased Estimator
What property does a BLUE mean estimator have?
It has the lowest variance among all linear unbiased estimators.
In the sample mean estimator, what are the weights (π€π)?
π€π = 1/n
What does the Law of Large Numbers (LLN) establish?
It establishes the large sample behavior of the mean estimator.
What is the Kolmogorov Strong Law of Large Numbers?
It states that the average of πππ random variables converges almost surely to the expected value.
What is the significance of consistency in estimators?
An estimator is consistent if its finite sample bias diminishes as sample size increases.
What happens to the variance of a consistent estimator as sample size increases?
The variance converges to zero.
True or False: The sample mean is normally distributed for large sample sizes when data are πππ normally distributed.
True
What does the Central Limit Theorem (CLT) state about the sample mean estimator?
It states that the sample mean estimator is approximately normally distributed for large samples.
What is the formula for the sample covariance estimator?
π^πF = (1/(π - 1)) β(Xπ β πΜπ)(ππ β πΜF)
What is the relationship between covariance and correlation?
Correlation is the standardized version of covariance.
How is the correlation coefficient estimated?
π^πF = π^πF / (π^π * π^F)
What is the median in a distribution?
The 50% quantile where probabilities of values above or below it are equal.
How is the median estimated from sorted data when the sample size is odd?
median(x) = x[n + 1/2]
What is the formula for estimating the interquartile range (IQR)?
IQR = q^75 - q^25
Why are quantiles considered robust to outliers?
They are unaffected by extreme values, unlike the mean.
What does the CLT imply about the distribution of the sample mean?
It implies that the distribution of the sample mean is centered on the population mean and variance declines as sample size increases.
What is the definition of covariance?
Covariance measures the linear dependence between two random variables.
What does the term βalmost surelyβ mean in the context of convergence?
It indicates that the convergence occurs with probability 1.
What happens to the distribution of the sample mean as sample size increases?
The distribution becomes narrower and more concentrated around the population mean.
What is the relationship between the sample means of two variables and the CLT?
The CLT can be applied to the joint behavior of the two mean estimators as a bivariate statistic.
What is the significance of the 2-by-2 covariance matrix in the context of bivariate data?
It is used to describe the joint distribution of the mean estimators for two variables.
What does the bivariate Central Limit Theorem (CLT) imply about correlation in data?
Correlation in the data produces a correlation between the sample means, which is identical to the correlation between the data series.
Define coskewness in the context of random variables.
Coskewness measures the likelihood of one variable taking a large directional value whenever the other variable is large in magnitude.
How many different measures are there when computing cross p-th moments?
There are p - 1 different measures.
List the types of cross moments when applying the principle to the first four moments.
- No cross means
- One cross variance (covariance)
- Two measures of cross-skewness (coskewness)
- Three cross-kurtoses (cokurtosis)
What is the significance of coskewness being zero in a bivariate normal distribution?
The coskewness is always 0, indicating no sensitivity to the direction of one variable to the magnitude of the other.
Fill in the blank: The two coskewness measures are represented as _______.
[E s(X, X, Y), E X^2 Y - E X E Y]
What does cokurtosis measure in relation to two series?
Cokurtosis captures the sensitivity of the magnitude of one series to the magnitude of the other series.
What are the configurations of the three measures of cokurtosis?
- (1,3)
- (2,2)
- (3,1)
True or False: The (2,2) cokurtosis measure is the easiest to understand.
True
How is the (2,2) cokurtosis measure defined?
It is defined as capturing the sensitivity of the magnitude of one series to the magnitude of the other series.
What does a large (2,2) cokurtosis indicate?
It indicates that both series tend to be large in magnitude at the same time.
List the variables for which the sample analog of coskewness can be estimated.
- X
- Y
What is the formula for estimating coskewness?
sΜ X, X, Y = (1/n) β (X_i - ΞΌΜ_X)(Y - ΞΌΜ_Y)
What does the cokurtosis use in its definition?
It uses combinations of powers that add to 4.
List the variables involved in the cokurtosis measures.
- k(X, X, Y, Y)
- k(X, Y, Y, Y)
- k(X, X, X, Y)
What is the relationship between the sample means and their limiting distribution in practice?
Mean estimators are treated as if they are normally distributed.
What are the two measures of coskewness?
- E s(X, X, Y)
- E X^2 Y - E X E Y
What is the significance of the variance in the context of coskewness?
Coskewness is standardized by the variance of one of the variables and the standard deviation of the other.
Fill in the blank: The univariate skewness estimators are s(X, X, X) and _______.
s(Y, Y, Y)
